How to model an object with width and depth, based only on an image that shows the outline?
I thought it would be easily fixable but your shape is a bit more complexe than I thought. If I had to do it from scratch, I would follow the shape with some extrusions, like this:

If you want to start from your current mesh, delete the extrusion to only keep the edges, don't forget to remove doubles, give some extrusions to the remaining edges, like this:


Etc... then extrude to give it thickness and give it a Subdivision Surface modifier to smooth it.

You could also convert your edges to curves, and give it thickness, but it will also need rework.